Prob/Stat Seminar

We study stability under tensorization and projection-type operations of gradient-type estimates and other functional inequalities for Markov semigroups on metric spaces. With the help of some transportation-type inequalities obtained by Baudoin-Eldredge (EJP, 2021), we prove that the constants in the gradient estimates are independent of the dimension. Our results are applicable to hypoelliptic diffusions on sub-Riemannian manifolds and some hypocoercive diffusions. As a byproduct of our method, we obtain dimension- independent reverse Poincaré, reverse log-Sobolev, and gradient bounds for Lie groups with transverse symmetry and for non-isotropic Heisenberg groups. This is a joint work with Fabrice Baudoin (Arhus University) and Maria Gordina (University of Connecticut).

Every Cantor set can be realized as the boundary of a tree. In this talk we will describe a way of realizing any compact doubling metric space as the visual boundary of a Gromov hyperbolic graph, and describe a uniformization procedure that gives the compact metric space as the boundary of a uniform domain. This allows us to link Sobolev-type spaces with local energies on uniform domains to Besov spaces with nonlocal energies on the compact metric space.

In this talk we discuss the Hot Spots constant for bounded smooth domains that was recently introduced by S. Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use probabilistic techniques to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in R^d for both small and asymptotically large d that significantly improve upon the existing results. Moreover, we prove new bounds for the Hot Spots constant for Lipschitz domains on Riemannian manifolds with non-negative Ricci curvature. This is joint work with Hugo Panzo (St. Louis) and Jing Wang (Purdue).

The elastic random manifold serves as a paradigmatic example of an elastic interface suspended in a quenched disordered medium, with random polymers models being the most well-studied special case. Work in the physics community has suggested that the behavior of these models in high dimensions should enter a number of distinct phases, depending on the temperature, and structure of the disorder, with the famed depinning transition being the most studied.

We confirm a number of predictions for these phases, including existence, rigorously. The central result is a formula for the free energy in the high-dimensional limit, from which we are also able to obtain the behavior of a number of other statistics. The key tool in our computation involves adapting the multi-species synchronization method of Panchenko.

We consider state space models that combine ideas from functional data analysis and Fourier analysis. The models are originally motivated by questions raised in the context of mechanical ventilation in intensive care medicine. Analogous ideas can be applied to any time series where non-constant time-dependent periodic patterns are expected. Asymptotic statistical inference and prediction are based on functional limit theorems, with weak convergence in suitable Hilbert spaces of sequences. This is joint work with Jeremy Naescher and Stefan Walterspacher.

Spin glasses are disordered statistical physics system with both ferromagnetic and anti-ferromagnetic spin interactions. The Gibbs measure belongs to the exponential family with parameters, such as inverse temperature $\beta>0$ and external field $h\in R$. Given a sample from the Gibbs measure of a spin glass model, we study the problem of estimating system parameters. In 2007, Chatterjee first proved that under reasonable conditions, for spin glass models with $h=0$, the maximum pseudo-likelihood estimator for $\beta$ is $\sqrt{N}$-consistent. However, the approach has been restricted to the single parameter estimation setting. Despite efforts of many years, the joint estimation of $(\beta,h)$ for spin glasses has remained open. In this paper, under some easily verifiable conditions, we proved that the bi-variate maximum pseudo-likelihood estimator is jointly $\sqrt{N}$-consistent for a large collection of spin glasses, including the Sherrington-Kirkpatrick model and its diluted variants.

Based on a joint work with Wei-Kuo Chen and Arnab Sen.

We provide a systematic convergence analysis of the Hessian operator estimator from random samples supported on a low dimensional manifold. We show that the impact of nonuniform sampling, curvature, and boundary effect on the widely applied Hessian operator estimator is asymptotically negligible. In particular, our result justifies the key step in the algorithm Hessian Eigenmap for generic data manifolds, and clarifies the convergence rate of local quadratic regression on a Riemannian manifold. This is joint work with Hau-Tieng Wu (Courant Institute, NYU).